Asymptotic behavior of the prediction error for stationary sequences

IF 1.3 Q2 STATISTICS & PROBABILITY Probability Surveys Pub Date : 2022-10-12 DOI:10.1214/23-ps21
N. Babayan, M. Ginovyan
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引用次数: 1

Abstract

One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or nondeterministic) and on the dependence structure of the underlying observed process $X(t)$. In this paper we consider this problem both for deterministic and nondeterministic processes and survey some recent results. We focus on the less investigated case - deterministic processes. It turns out that for nondeterministic processes the asymptotic behavior of the prediction error is determined by the dependence structure of the observed process $X(t)$ and the differential properties of its spectral density $f$, while for deterministic processes it is determined by the geometric properties of the spectrum of $X(t)$ and singularities of its spectral density $f$.
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平稳序列预测误差的渐近性态
离散时间二阶平稳过程$X(t)$的预测理论中的一个主要问题是,当$n$变为无穷大时,在给定$X(t),$$-n\le t\le-1$的情况下,描述预测$X(0)$的最佳线性均方预测误差的渐近性态。这种行为取决于规律性(确定性或非确定性)和底层观察过程$X(t)$的依赖结构。在本文中,我们考虑了确定性和不确定性过程的这个问题,并综述了最近的一些结果。我们专注于研究较少的案例确定性过程。结果表明,对于非确定性过程,预测误差的渐近行为是由观测过程$X(t)$的依赖结构及其谱密度$f$的微分性质决定的,而对于确定性过程,它是由$X(t)$的谱的几何性质及其谱密度$f$的奇异性决定的。
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来源期刊
Probability Surveys
Probability Surveys STATISTICS & PROBABILITY-
CiteScore
4.70
自引率
0.00%
发文量
9
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